On bounded generalized Harish-Chandra modules (Q441303)

The authors study \(\mathfrak{k}\)-semisimple \((\mathfrak g,\mathfrak{k})\)-modules with bounded \(\mathfrak{k}\)-multiplicities, or otherwiseNEWLINENEWLINE\textit{bounded generalized Harish-Chandra modules}. They show that if there exists an infinite-dimensional simple bounded \((\mathfrak g,\mathfrak{k})\)-module, then \(r_{\mathfrak g}\leq b_{\mathfrak k}\) where \(r_{\mathfrak g}\) is half the dimension of a nilpotent \(\mathrm{ad}_{\mathfrak g}\)-orbit of minimal positive dimension and \(b_{\mathfrak k}\) is the dimension of a Borel subalgebra of \(\mathfrak{k}\). They also explicitly construct simple bounded generalized Harish-Chandra modules and as an application they provide a sufficient condition for a subalgebra \(\mathfrak{k}\subset \mathfrak g\) with \(r_{\mathfrak g}\leq b_{\mathfrak k}\) to have bounded multiplicities. As an application they classify all bounded reductive maximal subalgebras in \(\mathfrak{sl}_n\).